Topics
2. Quadratic Functions
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United States of AmericaTennessee
Integrated Math 2

2.02 Characteristics of quadratic functions

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Concept summary

A quadratic function is a polynomial function of degree 2. A quadratic function can be written in the form f(x)=ax^2+bx+c where a, b, and c are real numbers.

A parabola is the graph of a quadratic function.

From the graph of a quadratic function, we can identify key features including domain and range, x and y-intercepts, increasing and decreasing intervals, positive and negative intervals, and end behavior. The parabola also has the following two features that help us identify it, and that we can use when drawing the graph:

Axis of symmetry

A line that divides a figure into two parts, such that the reflection of either part across the line maps precisely onto the other part. For a parabola, the axis of symmetry is a vertical line passing through the vertex.

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Vertex

The point where the parabola crosses the axis of symmetry. The vertex is either a maximum or minimum on the parabola.

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We can determine the key features of a quadratic function from its graph:

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This is a graph of the parent quadratic function: f(x)=x^2

  • Axis of symmetry: x=0
  • Vertex: \left(0,0 \right)
  • y-intercept at \left(0, 0\right)
  • x-intercept at \left(0, 0\right)
  • Domain: \left\{x \middle\vert - \infty \lt x\lt \infty\right\}
  • Range: \left\{y \middle\vert y\geq 0\right\}
  • As x \to \infty, f(x) \to \infty
  • As x \to - \infty, f(x) \to \infty

Worked examples

Example 1

Graph the quadratic function: f(x)=x^2-2x+1

Approach

We can create a table of values that satisfy f(x) and use it to help graph the function. It can be useful to choose values for x that are positive and negative, as well as x=0:

x-2-101234
f(x)

To complete the table, evaluating the function for each x-value.

Here is how we can obtain f(-2):

f(-2)=(-2)^2-2(-2)+1

f(-2)=4+4+1

f(-2)=9

Repeat this process for each x-value in the table.

x-2-1-01234
f(x)9410149

Now we can use these points to graph the quadratic function f(x)

Solution

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Reflection

Having the vertex in your table is useful, since it tells you where the parabola will change direction. Sometimes the table values you select will not include the vertex of the function, depending on the quadratic function being graphed. If you plot your initial table values and find you are unsure where the parabola changes direction, you can add additional values to your table until you can identify where f(x) changes direction.

Example 2

Consider the graph of the quadratic function g(x):

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a

Determine the x and y-intercepts.

Approach

To find the x-intercepts, locate the places where the parabola crosses the x-axis.

To find the y-intercepts, locate the place where the parabola crosses the y-axis.

Solution

We can identify the intercepts on the graph:

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From the graph we can see the there are two x-intercepts at \left(-4, 0\right) and \left(2, 0 \right), and there is one y-intercept at (0,8).

b

Determine the domain and range.

Approach

To find the domain of g(x), we want to find all possible x-values for which g(x) could be graphed.

To find the range, we want to find all possible values of g(x). The vertex of a parabola affects the range of the function, as it will be the maximum or minimum value of g(x).

Solution

We can see that for a parabola, there are no restrictions on which x-values can be graphed as each side of the parabola continues infinitely in either x-direction.

This parabola opens down, so the y-value of the vertex is the maximum value of the function. The parabola continues infinitely in the negative y-direction.

Domain:- \infty < x < \infty

Range: - \infty < y \leq 9

c

Identify each interval where the function is either increasing or decreasing.

Approach

g(x) is increasing where the output values become larger as its input values become larger.

g(x) is decreasing where the output values become smaller as its input values become larger.

Parabolas change from increasing to decreasing or decreasing to increasing about the x-value of the vertex.

Solution

We can see from the graph of g(x) that the parabola has positive slope up until x=-1 and negative slope after x=-1.

The function is neither increasing or decreasing at the vertex.

Increasing: x < -1

Decreasing: x > -1

d

Identify each interval where the function is either positive or negative.

Approach

In order to find the intervals for where the function is positive, we want to find the set of x-values where the parabola lies above the x-axis.

Similarly, to find the intervals where the function is negative, we want to find the set of x-values where the parabola lies below the x-axis.

The x-intercepts are where the function crosses the x-axis, in other words where the function is equal to 0. These values will help us determine where the function is positive and negative.

Solution

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We can see that the parabola lies above the x-axis between x=-4 and x=2. We can also see that the parabola lies below the x to the left of x=-4 and to the right of x=2.

Positive: -4 < x < 2

Negative: x < -4, x > 2

e

State the end behavior of the function.

Approach

In order to find the end behavior, we want to find the y-value that the function approaches as the x-values approach positive and negative infinity. We can see on the graph that as the x-values on the parabola get larger in either direction, the parabola continues downwards, or towards - \infty.

Solution

x \to \infty, y \to - \infty

x \to - \infty, y \to - \infty

Example 3

The graph shows the height of a softball above ground (in ft.) x seconds after it was thrown in the air.

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a

Find the y-intercept and describe what it means in context.

Approach

We want to find the place where the parabola crosses the y-axis.

Once we find the y-intercept, we want to connect this to the context of the softball. Since the y-axis represents the height of the softball in feet above ground, we can use it to identify the height of the softball at 0 seconds.

Solution

We can identify the y-intercept on the graph:

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The y-intercept is \left(0, 6\right)

The y-intercept tells us that the softball was thrown from a height of 6 feet above the ground.

b

Find the value of the x-intercept and describe what it means in context.

Approach

We want to find the place where the parabola crosses the x-axis.

Once we find the x-intercept, we want to connect this to the context of the softball. Since the x-axis represents the time in seconds after being thrown, we can use it to identify at how many seconds does the softball hit the ground.

Solution

We can identify the x-intercept on the graph:

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The x-intercept is \left(3, 0\right)

The x-intercept tells us that the softball hits the ground 3 seconds after it was thrown in the air.

c

Find the value of the vertex and describe what it means in context.

Approach

In order to find the vertex, we want to find the maximum point of the parabola.

Once we find the vertex, we want to connect this to the context of the softball. We know that the x-value of the vertex represents time in seconds after the softball is thrown and the y-value of the vertex represents the height of the softball above ground in feet.

Solution

We can identify the vertex on the graph:

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The vertex is \left(1.25, 12 \right)

After 1.25 seconds, the softball reaches a maximum height of 12 feet above ground.

Outcomes

M2.N.Q.A.1

Use units as a way to understand real-world problems.*

M2.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.

M2.F.IF.B.3

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.*

M2.F.IF.B.4

Relate the domain of a function to its graph and, where applicable, to the context of the function it models. *

M2.F.IF.C.6

Graph functions expressed algebraically and show key features of the graph by hand and using technology.*

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP2

Reason abstractly and quantitatively.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP4

Model with mathematics.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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